A pair of dice are rolled find the following, a) the probability of doubles, b) the odds in favor of a sum greater than 2, c) the probability of sum that is even and less than 5
To find the probabilities and odds in favor of certain events when rolling a pair of dice, we need to understand the sample space and the number of favorable outcomes for each event.
The sample space is the set of all possible outcomes when rolling two dice. Each die has six sides numbered 1 to 6, resulting in a total of 6 x 6 = 36 possible outcomes.
a) Probability of doubles:
Doubles are rolled when both dice show the same number, such as (1,1), (2,2), (3,3), and so on. There are six possible doubles outcomes (one for each number on a die). So, the probability of rolling doubles is 6/36, which can be simplified to 1/6.
b) Odds in favor of a sum greater than 2:
To find the odds in favor of a sum greater than 2, we need to determine the number of favorable outcomes for this event. A sum greater than 2 means that the dice must show any combination except (1,1). All other combinations result in sums greater than 2. There are 35 favorable outcomes (36 total outcomes minus the (1,1) combination). The odds in favor of this event can be expressed as the ratio of favorable outcomes to unfavorable outcomes. So, the odds in favor of a sum greater than 2 are 35:1.
c) Probability of a sum that is even and less than 5:
For the sum to be even and less than 5, we need to consider the possible outcomes: (1,1), (1,3), (3,1), (2,2), (1,2), and (2,1). There are 6 favorable outcomes. So, the probability of getting a sum that is even and less than 5 is 6/36, which simplifies to 1/6.
Remember, probability is a fraction that represents the likelihood of an event occurring, and odds describe the ratio of favorable outcomes to unfavorable outcomes.